Furrow Constriction in Animal Cell Cytokinesis Hervé Turlier, Basile Audoly, Jacques Prost, Jean-François Joanny  Biophysical Journal  Volume 106, Issue 1, Pages 114-123 (January 2014) DOI: 10.1016/j.bpj.2013.11.014 Copyright © 2014 Biophysical Society Terms and Conditions

Figure 1 (Left) Numerical cell shape and cortex thickness evolution. (t0) Initial spherical cortex of radius R0 and main ingredients of the model. The membrane is axisymmetric around the axis ez and is subjected to internal tensions Ns and Nφ in its axial t and azimuthal eφ principal directions, and to the cytoplasmic pressure P along its normal n. The acto-myosin layer of initial thickness e0 undergoes permanent turnover. Approximately 100 Lagrangian nodes are represented to follow the tangential membrane deformation over time (not all simulations nodes are shown). (t1, t2, and t3) Cell cortex snapshots at successive times of constriction, in response to the rescaled myosin activity signal ζ/ζmax illustrated by the color shading. The value rf is the furrow radius and Lp is the half pole-to-pole distance. Cortical flows along the membrane are represented by arrows of size proportional to the local tangent velocity. (Right) DIC microscopy images of a sand-dollar zygote (Dendraster) deprived of its hyaline layer and jelly coat at four equivalent times of furrow constriction. The cell is not flattened and scale bar is 20 μm. (Credits: G. Von Dassow.) To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 2 Furrow constriction dynamics. Time evolution of the furrow radius (rf = R0), and pole-to-pole half distance (Lp = R0). Numerical results (line) are compared to experimental measurements of a sand-dollar zygote (points, data from the same DIC microscopy images of Fig. 1). (Vertical dashed lines) Delimiters of the four phases of constriction described in the text and numbered from 0 to 3. (Inset) Equatorial signal I(t) applied as a function of time. To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 3 Constriction completion and failure. (A) Bifurcation diagram representing the final furrow radius rf∞/R0 as a function of the amplitude of equatorial overactivity δζ∞/ζ0. The diagram displays a jump from constriction failure to completion for a critical amplitude δζ∞/ζ0 ≈ 40. Final cell shapes are plotted for the six activity signals ζ, of amplitudes δζ∞/ζ0 = 10 (❶), 25 (❷), 40 (❸), 50 (❹), 75 (❺), and 100 (❻) as represented (inset) as a function of the contour length from equator s/L along the membrane (of length L). (Arrows) Hysteresis loop. Starting from a divided state above the threshold (❹, for example), we decrease the equatorial signal: the cell remains divided, unless the signal is dropped down to 0—the point at which it goes back to the spherical state. (B) Furrow radius evolution rf/R0 as a function of time t/Ta for the six signals ζ (represented in panel A, inset). To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 4 Influence of turnover on constriction. Furrow radius rf = R0 as a function of normalized time t = Ta for three turnover rates: kdTa = 30 (①), kdTa = 40 (②), and kdTa = 80 (③). (Inset) Corresponding steady-state membrane thickness e1 = e0 along the rescaled contour length s = L from equator (L is the total membrane midline length). To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 5 Cytokinesis duration is independent of initial cell size. Furrow radius rf as a function of time t = Ta for four initial cell radii R0 = 0.5, 1, 2, and 4. (Inset) Corresponding Gaussian activity signals of width w proportional to R0, plotted as a function of the membrane midline contour length s. To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 6 Scaling model. (A) Sketch of the minimal geometry proposed by Yoneda and Dan (53): Two portions of sphere of surface Ap and surface tension Na0 are pinched by an equatorial ring of radius rf, of width w, and of line tension γ. The opening angle θ characterizes the constriction state of the cell and the cytoplasmic volume-enclosed V0 is conserved. (B) Mechanical energy profile E = E0 as a function of the constriction state θ for four values of κ = γ/2R0Na0. Local minima of the energy correspond to equilibrium states (darker points), above which are plotted the corresponding cell shapes. (C) Bifurcation diagram representing the final furrow radius r∞f = R0 as a function of the control parameter κ. The upper branch and the branch rf = 0 are stable branches, but one branch (dot-dashed) is unstable. The critical point is a saddle-node, and the bifurcation classically exhibits an hysteresis (see arrows). Final cell shapes, starting from a spherical cell, are plotted for the six following values of the control parameter: κ = 0, 0.1, 0.25, 0.4, 0.5, and 0.75. To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions

Figure 7 Constriction dynamics in scaling. Normalized furrow radius evolution rf = R0 with time t = Ta. (A) For κ = 0.1, 0.25, 0.4, 0.5, 0.75, and 1, with constant λ = 0.1. For κ ≲ 0.4, the furrow radius reaches a plateau indicating constriction failure, whereas for κ ≳ 0.4 constriction is complete and its speed increases with κ. (B) For ef = e0 between 1 and 4, keeping ζf/ζ0 = 8 and w = R0 = 0.1 constant. Constriction slows down when ef = e0 decreases from 4 to 1.5, and can even fail when it drops to 1. (C) For four initial cell radii R0 = 0.5, 1, 2, and 4, where the ring width w is increased proportionally, w = 0.05, 0.1, 0.2, and 0.4. The values ef = e0 = 2 and ζf/ζ0 = 8 are maintained constant. The rate of constriction increases proportionally to the ring width, leading to the same constriction duration for the four cell sizes. (D) For a cell with dissipation due to the ring constriction only and with dissipation due to poles stretching and ring constriction (ζf/ζ0 = 8, ef = e0 = 2, w = R0 = 0.1). (Dashed line) Constriction of an isolated ring (no poles) fitted with the exponential function e−t/τ with τ = 2η/ζf Δμ. To see this figure in color, go online. Biophysical Journal 2014 106, 114-123DOI: (10.1016/j.bpj.2013.11.014) Copyright © 2014 Biophysical Society Terms and Conditions